non-equational physics derivations?

[gusl]

Do there exist physics derivations that make no use of equations or inequations?

Would algebraic derivations about symmetry groups (e.g. in gauge theories) be one such example?

Comments

[spoonless.livejournal.com]

Depends what you mean by equations I guess.

There are many derivations and manipulations done in physics which involve symbols without an equals sign. Although usually there is something similar to an equals sign, such as a proportionality symbol, or in the case of asymptotic series a ~, and also a ~ in the case where one expression is equivalent to another up to any finite amount added to either side (this is used a lot in quantum field theory). There is also derivations which involve no characters at all, just diagrams (for instance, Feynman diagrams).

And a lot of new principles or facts in physics are "proven"--or at least pointed out--in papers by using arguments rather than with equations. Although usually it would be possible to write something similar in equation form, as long as you specified what everything stood for.

Actually, come to think of it all of classical mechanics and quantum field theory is done from a starting point which doesn't involve any equal points. Everything is derived from the Lagrangian, but the "equations of motion" are not equal to anything in the Lagrangian, they are just found by taking various derivatives of it and setting them equal to each other. So in order to do the first step, you always need an input that isn't related to the next step by an equal sign.

I'm sure there are a ton of more examples... but I'm not thinking of them off the top of my head and this is probably enough for now to give you the idea that... while a lot of physics makes use of equals signs, it's by no means universal.

[gustavolacerda.livejournal.com]

But can't you always translate a derivation (including Feynman diagrams, etc) into equations?

[gaspaheangea.livejournal.com]

I hear topological quantum field theory is all category theoretic, where equations are replaced by isomorphisms.

[smandal.livejournal.com]

If by equations you mean equivalence relations, then no.

If you mean "=" then yes, all the time.

[gustavolacerda.livejournal.com]

Can you give one example, where you have these equivalence relations, and they couldn't be translated into equations?

I can understand why you'd do something without equations if you're doing meta-physics in order to unify different areas of physics (not to be confused with "metaphysics").

[smandal.livejournal.com]

The most obvious examples are symmetry transformations (e.g., electromagnetic gauge), and at a higher level dualities between whole theories (e.g., AdS/CFT correspondence).

[gustavolacerda.livejournal.com]

"conformal field theory"
"anti-de siter theory"

I'll have to look them up later.

[smandal.livejournal.com]

Briefly, that the field theory on a hypersurface can encode the data in a bulk gravitational theory it borders.

Another example: totally different systems such as liquid-gas states and ferromagnetic spin domains have the same critical exponents approaching phase transitions. There's no way to tell this by looking at a specific system, but the framework of renormalization groups can predict the right values based on a few features such as symmetry and dimension. So we say that various thermal systems are in the same universality class -- they're not the same microscopically, but they have the same relevant structure on this aspect of their behavior.

In a sense, all of physics is like this. We identify qualia (e.g., momentum, mass, etc.) by the equational relationships they exhibit -- sort of a Saussurean view. Electrical charge is an obvious example. So the science is in proposing an equivalence, then testing the equations.